On certain functional equation in semiprime rings and standard operator algebras (Q2877839)

The author proves the following result: Let \(\mathcal L(X)\) be the algebra of all bounded linear operators on a real or complex Banach space \(X\) and let \(\mathcal A(X)\subseteq \mathcal L(X)\) be a standard operator algebra. If a linear mapping \(D:\mathcal A(X)\to \mathcal L(X)\) satisfies NEWLINE\[NEWLINE2D(A^n)=D(A^{n-1})A+A^{n-1}D(A)+D(A)A^{n-1}+AD(A^{n-1})NEWLINE\]NEWLINE for all \(A\in \mathcal A(X)\) and some fixed positive integer \(n>2\), then \(D\) is a linear derivation. In particular, \(D\) is continuous.